Question

Please, look at the photo and give the ansver.

Answer 1

`\int \limits_1^(\infty)(\ln x)/(x^2)\,dx=\lim_(t\to\infty)\int \limits_1^t (\ln x)/(x^2)\\\\ \int (\ln x)/(x^2)\,dx=(*)\\ u=\ln x,du=(1)/(x) \\ dv=(1)/(x^2),v=-(1)/(x)\\ (*)=\ln x\cdot(-(1)/(x))-\int ((1)/(x)\cdot(-(1)/(x)))\, dx=\\ -(\ln x)/(x)+\int(1)/(x^2)\, dx=\\ -(\ln x)/(x)-(1)/(x)+C\\\\ \lim_(t\to\infty)\int \limits_1^t (\ln x)/(x^2)=\lim_(t\to \infty)\left[-(\ln x)/(x)-(1)/(x) \right]_1^t=`

`\lim_(t\to \infty)\left(-(\ln t)/(t)-(1)/(t)-\left(-(\ln 1)/(1)-(1)/(1)\right)\right)=\\ \lim_(t\to \infty)\left(-((\ln t)')/(t')\right)-0-(-1)=\\ \lim_(t\to \infty)\left(-((1)/(t))/(1)\right)+1=\\ \lim_(t\to \infty)\left(-(1)/(t)\right)+1=\\ 0+1=\\ 1`